### Numerical simulation

To confirm the validity of the method proposed in this article, we simulated a set of one-dimensional signals with relatively high carrier frequency, as shown in Fig. 1a. The corresponding power spectrum of *I*_{1} is shown in Fig. 1b. The range of the selected + 1 order frequency spectrum is marked in the figure. In the expression of this signal, α = -0.3*x*^{2}, b=-0.3(*x* - 0.02)^{2} +0.9, the phase distribution *φ* = 0.1*x*^{2}, and the carrier frequency *k* = 6.6*π*. 3% random noise is added to this set of signals. The horizontal coordinates for this set of signals *x* ∈[–1.02, 0.90]; the corresponding *kx* ∈[–6.7*π*,5.94*π*]. The cutoff point of the signal is not selected to be at the position of a full cycle of the carrier frequency. The envelope is not centrally symmetrical. The phase shift between the two interference signals is δ = *π*/3. Sampling includes 201 points. The odd number for the sampling is for the easy search for central frequency under the EMFT method. The above described interference signal module is commonly seen in practice. The uneven background intensity is included in this pair of interferograms. The amplitude modulation differs at different positions of the interference fringes. We can see from Fig. 1a that this set of interference signals include about 6 fringes. We call it a “high” carrier frequency signal, in comparison with the signal described later in this article that includes only 0.5 fringes. It is relative in our discussion.

We move along the Y-axis the assessment function curve for the EMFT method in Fig. 2 for comparison with the assessment function for the method introduced in this article. As seen in the figure, the EMFT method can accurately capture the + 1 order when the carrier frequency is high. The global phase shift determined under the method in this article and under the EMFT method are respectively 1.044*rad* and 1.045*rad*. Both are consistent with the set value.

When we keep the other parameters unchanged, decrease the carrier frequency to *k* = 0.7*π*, and include approximately 0.35 fringe in the entire signal, the two interference signal curves are shown as in Fig. 3a. The corresponding power spectrum is shown as in Fig. 3b. In this scenario, the error will be greater than under EMFT if we perform calculation with the single point Fourier Transform method. Meanwhile, the EMFT method is useful to increase the precision of determination as it adjusts the location and the range of the integration based on different carrier frequencies, but its potential impacts on the assessment functions from inappropriately chosen windows are also evident. As the carrier frequency decreases, the + 1 order frequency spectrum is not readily determinable. We carefully select the range of the + 1 order frequency spectrum and mark it with a dotted square in Fig. 3b. Based on the + 1 order frequency spectrum range as shown in the figure, we use the EMFT method to calculate the assessment function and portray it, along with the result from the method from this article, in Fig. 4. As seen from the curves in the figure, the results from the two methods are patently different. The result from the method in this article is δ = 1.025*rad*, whereas the phase shift calculated from the EMFT assessment function is *δ* = 1.113*rad*. As our set phase shift value is *δ* = 1.047*rad*, the relative errors for the two methods are 2.1% and 6.3%. The comparison results show that the precision from the two methods are comparable if the fringes are relatively many; but as fringes are significantly fewer, the reported method is superior to the EMFT method in its functionality.

In order to further compare the relatively errors between the two methods, we gradually increase the carrier frequency of the simulated signal, i.e. to increase the number of the interference fringes, and perform calculation based on the two methods. We report the relatively error curves in Fig. 5. When the fringes are few, the calculation errors from both methods increase; as the fringes increase, the relative errors gradually decrease. Generally, the method in this article generates lower errors than EMFT; its curve depicting the change in error is rather flatter, showing that the error values are relatively stable. EMFT, in comparison with the method in this article, generates higher relatively errors; the fluctuation is also higher. The method introduced in this article is evidently more advantageous, particularly, when the fringes are rare.

From the perspective of signal shapes, the above described simulated signals include various interfering factors that may impact the calculation of global phase shifts, including, as documented in Literature ([21], 33–34), unevenness in background intensity and modulation envelopes (α ≠ constant, b ≠ constant), low carrier frequency, signal noises, non-periodic apodization, and etc. For a linear carrier interferogram, if the signal is non-periodic, the sideband of the carrier frequency signal will be broader; if the carrier frequency is not high enough, then it will spectrally aliased with the baseband signal. In practice, however, most interference signals are usually not single frequency. The location of apodization is not readily selected for reasons such as background and modulation unevenness.

### Experimental results

In this section, we will verify the performance of the algorithm with experiment data. A group of linear carrier interferograms with a Zygo GPI interferometer has been recorded. Fig. 6 shows these phase shift interferograms, the resolution of these images is 256 × 256 pixels. These interferograms do not have any pre-processed including filtering or background removal etc. For these interferograms, we perform calculation of data of each row and column between *M*_{1} (Fig. 6a) and *M*_{2} (Fig. 6b) with method mentioned in the previous section. The global phase shift range to try into the Eq.7 should be [0, 2*π*], but in order to avoid infinity, we choose a variable range of [0.03*π,* 1.94*π*] and search step is 0.0047*π*. And the corresponding assessment function values of all rows and columns are shown in Fig. 7a and b respectively.

Each column data in Fig. 7 shows the assessment function calculated from one slice data, x-axis indicates the phase step, and the color map means evaluation value. From the graph above, we can find the minimum point for each column in the graph, and display the result in Fig. 8. The solid line in blue and the dot line in red are represent estimated phase step from each row and column respectively. The RMS of two curves are 0.018*rad* and 0.028*rad*, while the average values are 1.597*rad* and 1.610*rad*. We use the average of the two averages as the final estimated global phase shift. The running time is 0.32*s* (mean of 10 independent runs) based on a computer with a 3.3GHz i5 CPU and 8GB RAM using MATLAB®.

Using the same method, we calculate the global phase step between *M*_{1} (Fig. 6a) and M_{3} (Fig. 6c), and plot the result in Fig. 9. The result indicates the phase step between these two frame is 3.14*rad*. From those three phase-shift interferograms, the sought phase distribution *ϕ*(*x*, *y*) can be extracted by the equations below.

$$ \left[\begin{array}{ccc}3& \sum \cos \left({\delta}_i\right)& \sum \sin \left({\delta}_i\right)\\ {}\sum \cos \left({\delta}_i\right)& \sum {\cos}^2\left({\delta}_i\right)& \sum \sin \left({\delta}_i\right)\cos \left({\delta}_i\right)\\ {}\sum \sin \left({\delta}_i\right)& \sum \sin \left({\delta}_i\right)\cos \left({\delta}_i\right)& \sum {\sin}^2\left({\delta}_i\right)\end{array}\right]\left[\begin{array}{c}{D}_1\\ {}{D}_2\\ {}{D}_3\end{array}\right]=\left[\begin{array}{c}\sum {M}_i\\ {}\sum {M}_i\cos \left({\delta}_i\right)\\ {}\sum {M}_i\sin \left({\delta}_i\right)\end{array}\right], $$

(8)

$$ \phi \left(x,y\right)=- arc\tan \left[\frac{D_3}{D_2}\right], $$

(9)

Where, *i* = 1,2,3, *δ*_{1} = 0, *δ*_{2}* =* 1.6037*rad*, *δ*_{3} = 3.146*rad*. The phase demodulation result for the considered experimental data is shown in Fig. 10. The linear carrier frequency pattern has been removed from the unwrapping map.